How does radio transmission with fast moving targets work (including LTE on phone), doesnt the doppler effect shift the frequencies of all radio waves?

Yup, it shifts but it's a minor shift and easily handled in the receiver. Receivers already need a little ability to tune the carrier frequency to account for ordinary variations in the circuits. From memory GPS's doppler shift is on the scale of a single digit khz, so Starlink's probably double that. A few khz of shift is no big deal for ghz carriers.